Question

How do you factor `f(x)=x^3-12x^2+36x-32` completely, given that (x-2) is a factor?

Answer 1

`x^3-12x^2+36x-32=(x-2)^2(x-8)`

Explanation:

Knowing that `(x-2)` is a factor you can divide the polynomial:

2.....|.......1.........-12.........+36.........-32
........|...................2..........-20.........+32
........|.......1.........-10..........+16...........0

and find:

`x^3-12x^2+36x-32=(x-2)(x^2-10x+16)`

We could use the quadratic formula to find the factors of `q(x)=(x^2-10x+16)`

or go simply by inspection:

1) q(x) has two changes in sign, so it has two positive real roots.
2) The product of the roots is `16`
3) the sum of the roots is `10`

It's easy to see that `x=2` and `x=8` are such roots, and thus:

`x^3-12x^2+36x-32=(x-2)^2(x-8)`